English
Re-states exists_of_le in a slightly different binder arrangement.
Русский
Повторно формулируется exists_of_le в иной связке переменных.
LaTeX
$$Exists fun y => And (Eq x y) (Eq (T x) (S y))$$
Lean4
instance semilatticeInf : SemilatticeInf (E →ₗ.[R] F)
where
le := (· ≤ ·)
le_refl f := ⟨le_refl f.domain, fun _ _ h => Subtype.eq h ▸ rfl⟩
le_trans := fun _ _ _ ⟨fg_le, fg_eq⟩ ⟨gh_le, gh_eq⟩ =>
⟨le_trans fg_le gh_le, fun x _ hxz =>
have hxy : (x : E) = inclusion fg_le x := rfl
(fg_eq hxy).trans (gh_eq <| hxy.symm.trans hxz)⟩
le_antisymm _ _ fg gf := eq_of_le_of_domain_eq fg (le_antisymm fg.1 gf.1)
inf f g := ⟨f.eqLocus g, f.toFun.comp <| inclusion fun _x hx => hx.fst⟩
le_inf := by
intro f g h ⟨fg_le, fg_eq⟩ ⟨fh_le, fh_eq⟩
exact
⟨fun x hx => ⟨fg_le hx, fh_le hx, (fg_eq (x := ⟨x, hx⟩) rfl).symm.trans (fh_eq rfl)⟩, fun x ⟨y, yg, hy⟩ h =>
fg_eq h⟩
inf_le_left f _ := ⟨fun _ hx => hx.fst, fun _ _ h => congr_arg f <| Subtype.eq <| h⟩
inf_le_right _ g := ⟨fun _ hx => hx.snd.fst, fun ⟨_, _, _, hx⟩ _ h => hx.trans <| congr_arg g <| Subtype.eq <| h⟩
